1. Field of the Invention
This invention relates to an art of transforming image data using a Wavelet transform, and more particularly to an apparatus for compressing image which compresses any size of image data using a Wavelet transform as well as about an apparatus for restoring image which restores the image data compressed by that image compressing apparatus.
2. Description of the Related Art
In recent years, with growing diffusion of computers and communication networks, opportunities for collecting, sending and receiving a large quantity of image data have been increasing. With that reason, the technology of efficient compressing image data has been demanded.
Images to be subject of this invention include many things from human figure images, landscape pictures, computer graphics to medical data, weather data and astronomical observation data.
With image compressing systems like JPEG, the images have been divided into small block units to be processed by the block unit, assigned zero to the fractions.
When a Wavelet transform system is used, however, the whole image is collectively transformed since the computation efficiency will be rather drooped by dividing into the blocks. In that case, one side length of the subject image must be only an exponentiation of 2, allowing only using images with the 64.times.64, 256.times.256 pixels, etc. before.
There are the literature cited for the Wavelet transform as follows;
I. Daubechies "Ten lectures on Wavelets" 1992 SIAM (Society for Industrial and Applied Mathematics).
W. H. Press et. al "Numeral Recipes (Second Edition)" 1992 Cambridge University Press pp. 584-599.
FIG. 1a-FIG. 1c describe the compression processing using a Wavelet transform. To make it simple, the original image size will be described as 8.times.8 pixels (cf. the FIG. 1a) and to x and y directions, 3 octaves of multiple resolution analysis which will be described later will be conducted.
In the Wavelet transform, the frequency components will be divided into low region components and high region components for each horizontal and vertical directions. With it, the image data will be transformed as follows; information of low region components of horizontal direction and low region components of vertical direction in the up left of the image, information of high region components of horizontal direction and low region components of vertical direction in the up right, information of low region components of horizontal direction and high region components of vertical direction in the down left, information of high region components of horizontal direction and high region components of vertical direction in the down right. One transforming makes one octave of the resolution down.
Then, divide the part of low region components of horizontal direction and low region components of vertical direction in the up left of the image into the low region components and high region components of frequency in the same way. Repeat this several times.
Generally, in the image data, the information is concentrated in the low region components. Therefore, in the high region components, data reduction to some extent will not make much deterioration of the image by restoration. The Wavelet transform, which is a kind of orthogonal transform like DCT, has a merit that it has relatively less noise especially when the high region is transformed.
FIG. 1b describes the data after transform processing (multiple resolution analyzing processing). It shows that the low frequency components and high frequency components regions are lined up from the left to the right of x direction and from the up to the down of y direction of the transformed data, respectively. From this transformed data, it is clear that the closer the up left low frequency component region, the more the image data information is concentrated. When the data is reduced, use predetermined threshold value to make the data which is smaller than the threshold zero.
FIG. 1c shows the data after reduction processing. With it, the high frequency components region, which has low concentration of the information, will be efficiently reduced and then the efficient compressed data can be generated by compressing with encoding entropy, etc.
Restoration processing is conducted by inverse steps of the compression processing, which is, by decoding entropy and inverse Wavelet transforming to generate the restored data. FIG. 2 shows the data generated by the restoration processing.
As described above, when a Wavelet transform was used, one side length of the subject image must have been only an exponentiation of 2. If it was not an exponentiation of 2, the Wavelet transform should have been done after it had been extended to exponentiation of 2.
FIG. 3a and 3b describe extension processing of the prior Wavelet transform. In the prior Wavelet transform, only predetermined sized original image could be transformed. In case the size was 137.times.180 as shown in FIG. 3a, it was enlarged to the size 2.sup.8 .times.2.sup.8 =256.times.256 as shown in FIG. 3b and was transformed, using the extended image with zero value assigned to the enlarged area.
The multiple resolution analysis of a Wavelet transform is conducted as follows; Firstly, take out one line of x direction, then let brightness of the kth of the 256 pixels Ck.sup.(0). The value in the bracket of Ck.sup.(0) describes the transforming level. One octave down makes the value -1.
Next, following the resolving algorithm, obtain 128 low resolution components Cn.sup.(-1) and 128 Wavelet components dn.sup.(-1) using the following (expression 1) and (expression 2). EQU c.sub.n.sup.(-1) =(1/2).SIGMA..sub.k p.sub.k-n c.sub.k.sup.(0)(expression 1) EQU d.sub.n.sup.(-1) =(1/2).SIGMA..sub.k q.sub.k-2n c.sub.k.sup.(0)(expression 2)
.SIGMA..sub.k indicates the sum total from k=0 to 255. Also, p.sub.k and q.sub.k are coefficients of two scale functions; a scaling function .phi.(x) and a Wavelet function .psi.(x), respectively. EQU .phi.(x)=.SIGMA.p.sub.k .phi.(2x-k) (expression 3) EQU .psi.(x)=.SIGMA.q.sub.k .phi.(2x-k) (expression 4)
However, .SIGMA. indicates the sum total from k=0 to m (support length of a function: length of the area which function value is not zero.)
After finishing the above resolution to all the x direction lines, conduct the same to all the y direction rows as well. As a result, the low resolution components will be gathered in the 128.times.128 area.
Next, again, conduct the same to the 128.times.128 area to get low resolution component of 64.times.64. Repeat this until it gets the predetermined resolution. Normally around 4 times would be enough.
Generally, in compression using transforming, the higher the compressibility ratio is, the lower the quality of the restored image. In case of JPEG, it has guaranteed the quality of image with establishing "Quality Parameter".
This method, however, is not accurate since it reduces the data by adjusting the divided value with an integer. In this case, the signal-to-noise ratio was calculated after output of the restored image, compared with the original image.
As stated above, the image compression by a Wavelet transform for any sized image has not been studied till now in spite of its necessity as a practical matter. Processing process must be simple with the method of assigning zero value to the extended area by extending the image frame like existing JPEG system. However, the computing time and the amount of memory needed will be increased with the increasing amount of the assigned zero. If things come to the worst, four times of the amount of original memory would be needed.
Also, it took a long time to check the restored image quality since it had to compare the original image and restored image after restoring the image.